Find the least common multiple $(\text{LCM})$ of $2x^4-6x^3-8x^2$ and $4x^3-4x$. You can give your answer in its factored form.
Answer: The least common multiple $(\text{LCM})$ of two polynomial expressions is the polynomial with the least number of factors that is divisible by both polynomials. [How does this relate to the least common multiple of integers?] We can find the $\text{LCM}$ by factoring the two polynomials as much as possible and then comparing the factors: $2x^4-6x^3-8x^2$ can be factored as ${(2)(x)}{(x)}{(x+1)}{(x-4)}$ by factoring out a $2x^2$ and using the sum-product pattern. $4x^3-4x$ can be factored as ${(2)}{(2)(x)(x+1)}{(x-1)}$ by factoring out a $4x$ and using the difference of squares pattern. We can see that: Both polynomials share the factors ${(2)(x)(x+1)}$ Only the first polynomial has the factors ${(x)(x-4)}$ Only the second polynomial has the factors ${(2)(x-1)}$ Therefore, the least common multiple is the product of all the above factors: [Why?] $\begin{aligned}&\phantom{=}{(2)(x)(x+1)}{(x)(x-4)}{(2)(x-1)}\\\\ &=4x^2(x+1)(x-1)(x-4)\end{aligned}$ In conclusion, the least common multiple of the two polynomials is $4x^2(x+1)(x-1)(x-4)$.